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Why aren't leading-edge root extensions used on airliners?

Wouldn't they result in shorter wings and smaller slats?

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Leading edge extensions like slats are used in airliners. Most of these are controlled ones, so as to improve low speed characteristics.

A-319

By Adrian Pingstone (Arpingstone) - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=4761104

The dog tooth extensions were used in some older airliners like the Ilyushin Il-62 and the Vickers Super VC-10, he reason being the requirement to reduce drag.

Il- 62

By Sergey Lysenko - File:Air-to-air with a Russian Air Force Ilyushin Il-62M.jpg on Wikimedia Commons, originally published on http://russianplanes.net/id164205, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=40618026

The other leading edge extensions like LERX (and even dog tooth) are mostly found in high performance combat aircraft as these increase the stall angle and improve maneuverability; these are not the typical concerns in the commercial airliners. In airliners, vortex generators are used for preventing separation at low speed (high aoa).

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